Mathematical interpolation has been used for many years in many fields of science for approximating curves between known signal sample points. The subject of interpolation is treated in many text-books, such as Robert L. Ketter and Sherwood P. Prawel Jr., entitled "Modern Methods of Engineering Computation", McGraw Hill 1969. Many advanced methods of interpolation have been used in digital signal processing for many years, as described in Rabiner L. R. and Schafer "Digital Processing of Speech Signals", Prentice Hall, 1978, as well as George Wolberg entitled "Digital Image Warping", IEEE Computer Society Press, 1988.
According to the known prior art interpolation techniques utilized in digital signal processing, interpolation has been restricted to either linear interpolation or cubic interpolation. Quadratic interpolation has not been hitherto implemented in digital signal processing systems because prior art quadratic interpolators are known to be space variant and to introduce phase distortion in the output signal. One paper that is often cited as being the definitive paper on this subject is by R. W. Schafer and L. R. Rabiner and is entitled "A Digital Signal Processing Approach to Interpolation", Proceedings of the IEEE, volume 61, June 1973.
Linear interpolation is a commonly used form of interpolation in which a straight line EQU y=ax+b (equation 1)
is passed through two known points (e.g. y(0) and y(1)). Points that lay between x=0 and x=1 are determined by evaluating EQU y=y(0)+(y(1)-y(0))x for 0.ltoreq.x.ltoreq.1. (equation 2)
Linear interpolation has been preferred over cubic interpolation in the field of digital signal processing since, even though the frequency response of linear interpolation is inferior to that of cubic interpolation. Cubic interpolation systems require a great deal more calculation and storage hardware to implement than linear interpolation.
Quadratic interpolation as described in the above-discussed reference of Ketter and Prawel, and many others, is achieved by fitting a quadratic equation of the form EQU y=ax.sup.2 +bx+c (equation 3)
to three known points (e.g. y(-1), y(0) and y(1)). The resulting interpolation equation is: EQU y=((y(-1)-2y(0)+y(1))/2)x.sup.2 +((y(1)-y(-1))/2)x+y(0)-1.ltoreq.x.ltoreq.+1 (equation 4)
The main disadvantage of using quadratic interpolation in digital signal processing and other applications is the distortion which ocurrs at the transition from one set of sample points to another. As discussed in greater detail below with reference to the Figures, incorrect selection of a reference sample point can result in significant curve discontinuities (i.e. distortion) in the output interpolated curve approximation between the points.